Euler Line and the Nine-Point Center Construct ΔABC and the following nine points (continu...

Euler Line and the Nine-Point Center Construct ΔABC and the following nine points (continuing from the construction of Problem 12): midpoints of the sides (D, E, I), feet of the altitudes (J, K, L), and the Euler points (M, N, O) (the midpoints of segments , and ). Does it appear that these nine points lie on a circle? Construct the circumcenter U ( = R) of ΔDEI by finding midpoints P and Q of segments and perpendiculars at these points. Hide objects of construction.

(a) Does it appear that U lies on the Euler Line of points O, G, and H?

(b) Measure the distances HU(= HR), UO(= RF), and HO( = HF). Did you discover anything else peculiar? State a general theorem concerning the points H, U, G, and O (orthocenter, Nine-Point Center, centroid, and circumcenter) of any triangle ΔABC.

NOTE: Save the Euler segment and the four points H, U, G, and O for the next problem.